Intuition behind the Banach fixed-point theorem

Question

The theorem appeared as an exercise in my real analysis book and only considered functions in $\mathbb{R}$ but the proof of the general theorem seems to be almost identical after looking up the wikipedia article. Is there a somewhat intuitive way of thinking about this theorem? I understand the proof and what it entails but I don't see why the result ought to hold given the necessary conditions. Is there a way of convincing someone that the theorem ought to be true without actually proving it? Some vague geometric intuition would be nice to have a picture in my head of what's going on.

Answer 1

Generally, a function between metric spaces can be very wild with respect to the distances. But if we demand that the function actually decreases the distance between any two points, then it becomes difficult to actually construct such functions. Looking at a few examples of such functions one sees that one easy way to obtain such a function from a space to itself is to choose a point and treat it as a sort of magnet, where the function describes how points move toward it, as if the point exerts a gravitaional field, thus shrinking distances. If the gravitational pull is strong enough, the shrinking will be at least by a factor smaller than 1. The Banach fixed point theorem says that any endo-function shrinking distances sufficiently fast must be the result of such a point pulling things to it. If the space is not complete then that point may not be in the space but rather in its completion, but it’s still ‘there’. So, if globally the function behaves like the result of a pulling point with sufficient force, then that point is really there, since (intuitively?) if there is no point pulling everything to it, how can all the distances be tapidly shrinking no matter where you are?

Answer 2

One could image a map of let's say the state Wyoming lying somewhere in Wyoming (it looks nicely square-shaped, but you could choose any country/state). The Banach fixed point theorem then says that exactly one point on this map lies exactly on the corresponding point in the state.

The state is non-empty obviously, the metric is the Euclidean metric in $\mathbb{R}^2$, the state is obviously bounded (not really needed here), and we assume that the boundaries of the state are part of the state, so it is a closed space (I mean closed as a subset of the complete space $\mathbb{R}^2$), and therefore complete. Since Wyoming is our chosen space, the map must lie fully in this space (so not half outside Wyoming). Our contracting mapping is just the scale of the map. Now we have that the Banach fixed point theorem holds.

Let's say the state were an open set (so the space is not complete), then we could place the map such that the left boundary of the map lies on the left boundary of the state. Then there is no fixed point (the fixed point would be on the boundary, but that is not in the space now).

Answer 3

Suppose my disk is a portion of the Washington DC area, say the part inside or on the circular beltway. I also bring a map of the region, drawn on a piece of paper $P$. My map is thus a continuous map $D \to P$. If I am so callous as to crumple up the map and throw it out of the car window, so that it lands inside the beltway, I get an additional continuous map $P$ .$D$ ($p$ for ‘projection’), assigning to each point on the crumpled paper the point on the ground directly under it. Brouwer’s theorem, applied to the map $f =p \circ m: D \to D$, tells me that some point x inside the beltway is directly under the point $m(x)$ that represents x on the map. Do you find that surprising? I did when I first heard it. You can try the experiment, but please pick up the map afterward.

Session-10, page-121, Conceptual Mathematics by Lawere,Scahnuel